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Correct Calculation Of Percentage Error


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    Sometimes your computer may display an error code indicating that an error percentage is being calculated. There can be several reasons for this error to appear. Computing a partial error involves using the total error, which is simply the difference between the observed value and the true value. The absolute error is then divided by the true value, which gives the relative error, which is multiplied by 100 to get a certain percentage error.

  • Differences
  • Linear fit
  • Error propagation
  • Think about the differences by breaking down the “fraction” that (displaystyle fracdydx) extracted when we differentiated the function.
    < br>We found out where the derivative or rate change of any function can be displayed as (displaystyle fracdydx=f’left( x right)) where (dy) is infinitesimal, change to (y), and (dx ) (or (Delta x)) is an infinitesimal change to (x). It turns out that if (fleft( x right)) is completedifferentiable function on a separate open interval containing (x) and your current differential (x ) (( dx ) ) is a non-zero real number, then (dy=f’left( a right)dx ) (see how we just increased both parts of (dx))? And I won’t go into details at a specific point, but the differential associated with (y) can be used to approximate the nature of the change in (y), i.e. ( Delta y approximately dy ). (I always forget, but it’s important to remember that (Delta y=fleft( x+Delta x right)-fleft( x right)).)

    Computing Difference

    We’ve learned the rules of differentiation before, and most of them apply to differentials as well. Sounds familiar, right? Note that many of us have to apply the differential delivery rule in this problem.

    We can also use differentials to perform linear feature evaluations (we did it here with the tangent approximation) with this formula, which is similar to the brand new point-slope formula (remember, the mark is the slope). : (y-y_0=f’left( right)left( x_0 x-x_0 right)), or (fleft( back button right)-fleft( x_0 right) = f’left( right)left( x_0 x-x_0 right)), which means a (fleft( x right)=fleft( x_0 right)+f’ left ( right) left( x_0 x-x_0 right)). And remember that variables and index “0” are “old” values. Consider the equation as basic (y)”, “new equals” old (y)”, preferably derived from “old (x)”. situations, there is no doubt the difference between “new (x)” and “old (x)”.

    (And remember that we’re solving problems like this, so we’re going to “enjoy math”, that is, the way math is used, not calculators and computers.)

    Note. Another way to represent differentials is to use this formula; some coaches prefer this method: (displaystyle fracdydx=f’left( a right);,,dy=f’left( x right)dx) (this gives experience, n not A – is “up” and “up”). Once we get (dy), my partner and I just add it to the current original (y) to get the current approximation. This is also shown in the fourth problem below.

    percentage error calculus

    Here are a few cases where both differentials and feature scores are found:

    Problem Solution

    Find valueAn expression that combines (boldsymbol dy) and (boldsymbol Delta y) to form (x=4) and (Delta x=.1).

    (Remember that it could be (Delta y=fleft( x+Delta x right)-fleft( z right))))

    (The answers are obvious since (Delta x) is small)

    First find (dy) by simple differentiation:

    (displaystyle dx) y=x^2-1;,,,,fracdydx=2x;, , ,dy=,2xcdot If (x=4), then (Delta x=.1), .(displaystyle .dy=2left( .4 .right) , cdot ..1=.8).

    (displaystyle beginalignDelta y&=fleft( x+Delta x right)-fleft( by right)&=fleft( 4+.1 right)-f left( see right)&=left( 4.1^2-1 right)-left( 4^2-1 right)=.81endalign)

    Find some differences (dy) for:


    (displaystyle beginaligny&=4cosleft( 2x right)-8x^3fracdydx&=4cdot -sin left( 2x right)cdot 2-24x^2 dy&=left(-8sinleft( 2xright)-24x^2right)dxendalign)

    Use differentials and graph (f’left( a right)) (derivative) to approximate (fleft( 3.2 right) shown (f left( which is 3 right) =5).

    percentage error calculus

    Use a formula like this: (y-y_0=f’left( x_0 x-x_0 right)left( right))

    (I like to use this formula now because it looks like the slope of a dot. Remember that variables with index 0 are the “original” or “old” values ​​ofResponsibly). Note that this is a variant of the formula (fleft( right)=fleft( by x_0 right)+f’left( x_0 right)left( x-x_0 right) . ) .< /p>

    We identify this:

    (x_0) (y_0) (f’left( x_0 right)) (y) (x)
    3 5 2.25 ? 3.2

    So we get (y-y_0=f’left( x_0 right)left( x-x_0 Or right)), (y-5=2.25left( 3.2-3 right ) ). So (y=2.25left( 3.2-3 right)+5=5.45).

    Use differentials for approximation:


    Another way to get rid of the point-slope formula. Use (x) for 3 years ago, 4 for (y_0), (15–16=– 1) for (dx):

    (displaystyle beginalign&=sqrtx=x^frac12fracdydx&=frac12x^-frac12dy&=frac12x^-frac12dxdy&=frac12left( 16 right )^-frac12left( -1 right)=-frac18endalign)

    (displaystyle y=y_0+dy=4+-frac18=3.875)

    Use this formula: (y-y_0=f’left( x_0 right)left( x-x_0 right))

    The function is (y=sqrtx=x^frac12), so we have (displaystyle X f’left( right)=frac12x^-frac12). Now the trick is to directly find the simpler value in their function so that we can solve the device without a calculator. Let’s use (sqrt16=4). Now we have:

    (x_0) (y_0=sqrtx_0) (f’left ( x_0 right)) (y) (x)
    16 4 (frac12left( 16 right)^-frac12=.We 125) ? 15

    Then you have (y-y_0=f’left( x_0 right)left( x-x_0 right)) or (displaystyle y-4=.125left( 15-16 Right)). So (displaystyle y=0,125left( 15-16 right)+4=3,875).

    Compare this to what you get on your calculator. Pretty cool!

    Use differentials to approximate:

    (displaystyle sin left( 3 right))

    Use this formula again: (y-y_0=f’left( x_0 right)left( x-x_0 right))

    The part is (y=sin left( x right)), so we have (f’left( x right)=cos left( a right)) . Now the trick is to find the simpler dollar value in the function, so we might want to solve it without a calculator. Build (sinleft(pi right),,(pi approx 3.14) We now have:

    (x_0) (y_0=sinleft(x_0right)) (f’left( x_0 right)) (y) (x)
    (pi ) (sinleft(piright)=0) (cos left( pi right)=-1) ? 3

    Then we get (y-y_0=f’left( x_0 right)left( x-x_0 right)), (displaystyle y-0=-1left( 3- pi right)).


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  • Then (displaystyle y=pi -3=.14112).

    What is the percent error of the poll?

    Therefore, the percentage error calculation is as follows: percentage error = 14.29%. According to a public opinion poll conducted by the current news channel during the election campaign, to what extent did they count the XYZ party in terms of winning 278 seats out of 450.

    Wewe can use differentials in physics to evaluate trade-offs, for example, in physical sensors. For these problems, we usually take the most recent output and use “(dx)” for the “(dy)” part of the error output. Then, to get the error amount, divide the error by the total and multiply by 100.

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